本文研究了$\mathbb{R}^{3}$中非退化一维线族和平面族的投影。利用R. Kaufman，我们可以证明至多$1/2$维集合$B \subset \mathbb{R}^{3}$的Hausdorff维数通常在线上的一维投影族下保持不变。我们用一个$\varepsilon$改进了这个结果，证明了如果$\dim_{\mathrm{H}} B = s > 1/2$，那么投影的packing维数几乎必然至少是$σ(s) > 1/2$ .对于平面上的投影，我们得到了一个类似的界限，用$1$代替阈值$1/2$。在无旋转的自相似集$K \subset \mathbb{R}^{3}$的特殊情况下，得到了单参数直线投影族的完整的Marstrand型投影定理.这个结果的$\dim_{\mathrm{H}} K \leq 1$情形来自M. Hochman，但$\dim_{\mathrm{H}} K > 1$部分是新的：在这个假设下，我们证明了投影几乎必然有正的长度。
英文摘要：
We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most $1/2$-dimensional sets $B \subset \mathbb{R}^{3}$ is typically preserved under one-dimensional families of projections onto lines. We improve the result by an $\varepsilon$, proving that if $\dim_{\mathrm{H}} B = s > 1/2$, then the packing dimension of the projections is almost surely at least $σ(s) > 1/2$. For projections onto planes, we obtain a similar bound, with the threshold $1/2$ replaced by $1$. In the special case of self-similar sets $K \subset \mathbb{R}^{3}$ without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The $\dim_{\mathrm{H}} K \leq 1$ case of the result follows from recent work of M. Hochman, but the $\dim_{\mathrm{H}} K > 1$ part is new: with this assumption, we prove that the projections have positive length almost surely.
Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ''(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $l_θ=\{tγ(θ):t\in\mathbb{R}\}$ and $ρ_θ:\mathbb{R}^3\rightarrow l_θ$ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $A\subset\mathbb{R}^3$ and $0\le s\le 1$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(ρ_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+\frac{s-\text{dim}(A)}{2}$.
Marstrand's projection theorem from $1954$ states that if $K \subset \mathbb{R}^{3}$ is an analytic set, then, for $\mathcal{H}^{2}$ almost every $e \in S^{2}$, the orthogonal projection $π_{e}(K)$ of $K$ to the line spanned by $e$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. This paper contains the following sharper version of Marstrand's theorem. Let $V \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. Then, for $\mathcal{H}^{1}$ almost every $e \in S^{2} \cap V$, the projection $π_{e}(K)$ has Hausdorff dimension $\min\{\dim_{\mathrm{H}} K,1\}$. For $0 \leq t < \…
It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $θ\in [0,2π)$ the projection $π_θ(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos θ, \sin θ, 1)$ satisfies $\dim π_θ(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves the bound of Oberlin and Oberlin, and of Orponen and Venieri, for $\dim A \in (3/2,5/2)$. More generally, a weaker lower bound is given for families of planes in $\mathbb{R}^3$ parametrised by curves in $S^2$ with nonvanishing geodes…
